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Central Limit theorem and LLN

The Central Limit Theorem is one of the greatest results in probability theory because it says that the sum of a big number of variables has approximately a normal distribution.

We define a set of random variables iid X1, X2, X3, … , Xn with mean μ and variance σ². Then if n is very big, the sum

X1 + X2 + X3 + … + Xn

is approximately normal with mean and variance nσ². If we also normalize the sum we can say that

(X1 + X2 + X3 + … + Xn – nμ)/(σ√n)

is approximately a normal standard, so to a normal with mean 0 and variance 1.

Look for the most popular distributions in statistics

Distribution can be divided into two categories: discrete and continuous distributions.

The most popular discrete distributions are:

1)    Boolean (Bernoulli) which takes value 1 with probability p and value 0 with probability q = 1 − p.

2)    Binomial which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success

3)    Poisson which describes a very large number of individually unlikely events that happen in a certain time interval

4)    Hypergeometric which describes the number of successes in the first m of a series of n consecutive Yes/No experiments, if the total number of successes is known. This distribution arises when there is no replacement.

The most popular continuous distributions are:

1)    Normal (or Gaussian) often used in the natural and social sciences to represent real-valued random variables

2)    Chi-squared which is the sum of the squares of n independent Gaussian random variables. It is a special case of the Gamma distribution

3)    Gamma which describes the time until n consecutive rare random events occur in a process with no memory

4)    Beta a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities

5)    T-Student useful for estimating unknown means of Gaussian populations

6)    F-Distribution (Fisher) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance

7)    Weibull of which the exponential distribution is a special case is used to model the lifetime of technical devices and is used to describe the particle size distribution of particles generated by grinding, milling and crushing operations

 

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